LMFDB - Embedded newform 676.2.l.g.319.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [676,2,Mod(19,676)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(676, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 5]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("676.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 676 = 2^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	676.l (of order $12$, degree $4$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.39788717664$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label			319.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	676.319
Dual form			676.2.l.g.587.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.36603 - 0.366025i) q^{2} +(1.73205 - 1.00000i) q^{4} +(3.00000 + 3.00000i) q^{5} +(2.00000 - 2.00000i) q^{8} +(-1.50000 - 2.59808i) q^{9} +(5.19615 + 3.00000i) q^{10} +(2.00000 - 3.46410i) q^{16} +(-1.73205 + 1.00000i) q^{17} +(-3.00000 - 3.00000i) q^{18} +(8.19615 + 2.19615i) q^{20} +13.0000i q^{25} +(-2.00000 + 3.46410i) q^{29} +(1.46410 - 5.46410i) q^{32} +(-2.00000 + 2.00000i) q^{34} +(-5.19615 - 3.00000i) q^{36} +(-1.83013 - 6.83013i) q^{37} +12.0000 q^{40} +(-1.36603 + 0.366025i) q^{41} +(3.29423 - 12.2942i) q^{45} +(6.06218 + 3.50000i) q^{49} +(4.75833 + 17.7583i) q^{50} -14.0000 q^{53} +(-1.46410 + 5.46410i) q^{58} +(-5.00000 - 8.66025i) q^{61} -8.00000i q^{64} +(-2.00000 + 3.46410i) q^{68} +(-8.19615 - 2.19615i) q^{72} +(11.0000 - 11.0000i) q^{73} +(-5.00000 - 8.66025i) q^{74} +(16.3923 - 4.39230i) q^{80} +(-4.50000 + 7.79423i) q^{81} +(-1.73205 + 1.00000i) q^{82} +(-8.19615 - 2.19615i) q^{85} +(-1.09808 - 4.09808i) q^{89} -18.0000i q^{90} +(-1.83013 + 6.83013i) q^{97} +(9.56218 + 2.56218i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 12 q^{5} + 8 q^{8} - 6 q^{9} + 8 q^{16} - 12 q^{18} + 12 q^{20} - 8 q^{29} - 8 q^{32} - 8 q^{34} + 10 q^{37} + 48 q^{40} - 2 q^{41} - 18 q^{45} - 26 q^{50} - 56 q^{53} + 8 q^{58} - 20 q^{61}+ \cdots + 14 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 12 * q^5 + 8 * q^8 - 6 * q^9 + 8 * q^16 - 12 * q^18 + 12 * q^20 - 8 * q^29 - 8 * q^32 - 8 * q^34 + 10 * q^37 + 48 * q^40 - 2 * q^41 - 18 * q^45 - 26 * q^50 - 56 * q^53 + 8 * q^58 - 20 * q^61 - 8 * q^68 - 12 * q^72 + 44 * q^73 - 20 * q^74 + 24 * q^80 - 18 * q^81 - 12 * q^85 + 6 * q^89 + 10 * q^97 + 14 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/676\mathbb{Z}\right)^\times$.

$n$	$339$	$509$
$\chi(n)$	$-1$	$e\left(\frac{11}{12}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.36603	−	0.366025i	0.965926	−	0.258819i
$2$	1.36603	−	0.366025i	0.965926	−	0.258819i
$3$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$3$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$4$	1.73205	−	1.00000i	0.866025	−	0.500000i
$5$	3.00000	+	3.00000i	1.34164	+	1.34164i	0.894427	+	0.447214i	$0.147584\pi$
$5$	3.00000	+	3.00000i	1.34164	+	1.34164i	0.447214	+	0.894427i	$0.352416\pi$
$6$		0			0
$7$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$7$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$8$	2.00000	−	2.00000i	0.707107	−	0.707107i
$9$	−1.50000	−	2.59808i	−0.500000	−	0.866025i
$10$	5.19615	+	3.00000i	1.64317	+	0.948683i
$11$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$11$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$12$		0			0
$13$		0			0
$13$		0			0
$14$		0			0
$15$		0			0
$16$	2.00000	−	3.46410i	0.500000	−	0.866025i
$17$	−1.73205	+	1.00000i	−0.420084	+	0.242536i	−0.695113	−	0.718900i	$-0.744646\pi$
$17$	−1.73205	+	1.00000i	−0.420084	+	0.242536i	0.275029	+	0.961436i	$0.411312\pi$
$18$	−3.00000	−	3.00000i	−0.707107	−	0.707107i
$19$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$19$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$20$	8.19615	+	2.19615i	1.83272	+	0.491075i
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$			13.0000i			2.60000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.00000	+	3.46410i	−0.371391	+	0.643268i	−0.989780	−	0.142605i	$-0.954452\pi$
$29$	−2.00000	+	3.46410i	−0.371391	+	0.643268i	0.618389	+	0.785872i	$0.287786\pi$
$30$		0			0
$31$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$31$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$32$	1.46410	−	5.46410i	0.258819	−	0.965926i
$33$		0			0
$34$	−2.00000	+	2.00000i	−0.342997	+	0.342997i
$35$		0			0
$36$	−5.19615	−	3.00000i	−0.866025	−	0.500000i
$37$	−1.83013	−	6.83013i	−0.300871	−	1.12287i	−0.936442	−	0.350823i	$-0.885902\pi$
$37$	−1.83013	−	6.83013i	−0.300871	−	1.12287i	0.635571	−	0.772043i	$-0.280765\pi$
$38$		0			0
$39$		0			0
$40$	12.0000			1.89737
$41$	−1.36603	+	0.366025i	−0.213337	+	0.0571636i	−0.363905	−	0.931436i	$-0.618557\pi$
$41$	−1.36603	+	0.366025i	−0.213337	+	0.0571636i	0.150567	+	0.988600i	$0.451890\pi$
$42$		0			0
$43$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$43$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$44$		0			0
$45$	3.29423	−	12.2942i	0.491075	−	1.83272i
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$	6.06218	+	3.50000i	0.866025	+	0.500000i
$50$	4.75833	+	17.7583i	0.672930	+	2.51141i
$51$		0			0
$52$		0			0
$53$	−14.0000			−1.92305			−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000			−1.92305			−0.961524	+	0.274721i	$0.911414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−1.46410	+	5.46410i	−0.192246	+	0.717472i
$59$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$59$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$60$		0			0
$61$	−5.00000	−	8.66025i	−0.640184	−	1.10883i	−0.985391	−	0.170305i	$-0.945525\pi$
$61$	−5.00000	−	8.66025i	−0.640184	−	1.10883i	0.345207	−	0.938527i	$-0.387809\pi$
$62$		0			0
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$67$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$68$	−2.00000	+	3.46410i	−0.242536	+	0.420084i
$69$		0			0
$70$		0			0
$71$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$71$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$72$	−8.19615	−	2.19615i	−0.965926	−	0.258819i
$73$	11.0000	−	11.0000i	1.28745	−	1.28745i	0.351123	−	0.936329i	$-0.385800\pi$
$73$	11.0000	−	11.0000i	1.28745	−	1.28745i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	−5.00000	−	8.66025i	−0.581238	−	1.00673i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	16.3923	−	4.39230i	1.83272	−	0.491075i
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$	−1.73205	+	1.00000i	−0.191273	+	0.110432i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	−8.19615	−	2.19615i	−0.888998	−	0.238206i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.09808	−	4.09808i	−0.116396	−	0.434395i	0.882992	−	0.469389i	$-0.155526\pi$
$89$	−1.09808	−	4.09808i	−0.116396	−	0.434395i	−0.999388	+	0.0349934i	$0.988859\pi$
$90$		−	18.0000i		−	1.89737i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.83013	+	6.83013i	−0.185821	+	0.693494i	0.808632	+	0.588315i	$0.200208\pi$
$97$	−1.83013	+	6.83013i	−0.185821	+	0.693494i	−0.994453	+	0.105180i	$0.966458\pi$
$98$	9.56218	+	2.56218i	0.965926	+	0.258819i
$99$		0			0
$100$	13.0000	+	22.5167i	1.30000	+	2.25167i
$101$	−17.3205	−	10.0000i	−1.72345	−	0.995037i	−0.911479	−	0.411346i	$-0.865059\pi$
$101$	−17.3205	−	10.0000i	−1.72345	−	0.995037i	−0.811976	−	0.583691i	$-0.801608\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$	−19.1244	+	5.12436i	−1.85752	+	0.497721i
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$	7.00000	+	7.00000i	0.670478	+	0.670478i	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	7.00000	+	7.00000i	0.670478	+	0.670478i	−0.287348	+	0.957826i	$0.592774\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.00000	+	13.8564i	0.752577	+	1.30350i	0.946570	+	0.322498i	$0.104523\pi$
$113$	8.00000	+	13.8564i	0.752577	+	1.30350i	−0.193993	+	0.981003i	$0.562144\pi$
$114$		0			0
$115$		0			0
$116$			8.00000i			0.742781i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−9.52628	+	5.50000i	−0.866025	+	0.500000i
$122$	−10.0000	−	10.0000i	−0.905357	−	0.905357i
$123$		0			0
$124$		0			0
$125$	−24.0000	+	24.0000i	−2.14663	+	2.14663i
$126$		0			0
$127$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$127$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$128$	−2.92820	−	10.9282i	−0.258819	−	0.965926i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−1.46410	+	5.46410i	−0.125546	+	0.468543i
$137$	−9.56218	−	2.56218i	−0.816952	−	0.218902i	−0.173939	−	0.984757i	$-0.555649\pi$
$137$	−9.56218	−	2.56218i	−0.816952	−	0.218902i	−0.643013	+	0.765855i	$0.722316\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−12.0000			−1.00000
$145$	−16.3923	+	4.39230i	−1.36131	+	0.364761i
$146$	11.0000	−	19.0526i	0.910366	−	1.57680i
$147$		0			0
$148$	−10.0000	−	10.0000i	−0.821995	−	0.821995i
$149$	−6.22243	+	23.2224i	−0.509761	+	1.90246i	−0.0870170	+	0.996207i	$0.527733\pi$
$149$	−6.22243	+	23.2224i	−0.509761	+	1.90246i	−0.422744	+	0.906249i	$0.638933\pi$
$150$		0			0
$151$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$151$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$152$		0			0
$153$	5.19615	+	3.00000i	0.420084	+	0.242536i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	12.0000			0.957704			0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000			0.957704			0.478852	+	0.877896i	$0.341053\pi$
$158$		0			0
$159$		0			0
$160$	20.7846	−	12.0000i	1.64317	−	0.948683i
$161$		0			0
$162$	−3.29423	+	12.2942i	−0.258819	+	0.965926i
$163$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$163$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$164$	−2.00000	+	2.00000i	−0.156174	+	0.156174i
$165$		0			0
$166$		0			0
$167$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$167$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$168$		0			0
$169$		0			0
$170$	−12.0000			−0.920358
$171$		0			0
$172$		0			0
$173$	3.46410	−	2.00000i	0.263371	−	0.152057i	−0.362500	−	0.931984i	$-0.618077\pi$
$173$	3.46410	−	2.00000i	0.263371	−	0.152057i	0.625871	+	0.779926i	$0.284744\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−3.00000	−	5.19615i	−0.224860	−	0.389468i
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$	−6.58846	−	24.5885i	−0.491075	−	1.83272i
$181$		−	18.0000i		−	1.33793i	−0.743294	−	0.668965i	$-0.766738\pi$
$181$		−	18.0000i		−	1.33793i	0.743294	−	0.668965i	$-0.233262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	15.0000	−	25.9808i	1.10282	−	1.91014i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	−1.83013	−	6.83013i	−0.131735	−	0.491643i	0.868255	−	0.496119i	$-0.165242\pi$
$193$	−1.83013	−	6.83013i	−0.131735	−	0.491643i	−0.999990	+	0.00447566i	$0.998575\pi$
$194$			10.0000i			0.717958i
$195$		0			0
$196$	14.0000			1.00000
$197$	20.4904	−	5.49038i	1.45988	−	0.391173i	0.560431	−	0.828201i	$-0.310635\pi$
$197$	20.4904	−	5.49038i	1.45988	−	0.391173i	0.899448	+	0.437028i	$0.143969\pi$
$198$		0			0
$199$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$	26.0000	+	26.0000i	1.83848	+	1.83848i
$201$		0			0
$202$	−27.3205	−	7.32051i	−1.92226	−	0.515069i
$203$		0			0
$204$		0			0
$205$	−5.19615	−	3.00000i	−0.362915	−	0.209529i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$211$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$212$	−24.2487	+	14.0000i	−1.66541	+	0.961524i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	12.1244	+	7.00000i	0.821165	+	0.474100i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$223$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$224$		0			0
$225$	33.7750	−	19.5000i	2.25167	−	1.30000i
$226$	16.0000	+	16.0000i	1.06430	+	1.06430i
$227$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$227$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$228$		0			0
$229$	17.0000	−	17.0000i	1.12339	−	1.12339i	0.132164	−	0.991228i	$-0.457808\pi$
$229$	17.0000	−	17.0000i	1.12339	−	1.12339i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$		0			0
$231$		0			0
$232$	2.92820	+	10.9282i	0.192246	+	0.717472i
$233$		−	16.0000i		−	1.04819i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$		−	16.0000i		−	1.04819i	0.851658	−	0.524097i	$-0.175597\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	25.9545	+	6.95448i	1.67188	+	0.447978i	0.965615	−	0.259975i	$-0.0837143\pi$
$241$	25.9545	+	6.95448i	1.67188	+	0.447978i	0.706260	+	0.707953i	$0.250381\pi$
$242$	−11.0000	+	11.0000i	−0.707107	+	0.707107i
$243$		0			0
$244$	−17.3205	−	10.0000i	−1.10883	−	0.640184i
$245$	7.68653	+	28.6865i	0.491075	+	1.83272i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−24.0000	+	41.5692i	−1.51789	+	2.62907i
$251$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$251$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	27.7128	+	16.0000i	1.72868	+	0.998053i	0.895528	+	0.445005i	$0.146798\pi$
$257$	27.7128	+	16.0000i	1.72868	+	0.998053i	0.833150	+	0.553047i	$0.186535\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	12.0000			0.742781
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$	−42.0000	−	42.0000i	−2.58004	−	2.58004i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.0000	+	17.3205i	0.609711	+	1.05605i	0.991288	+	0.131713i	$0.0420477\pi$
$269$	10.0000	+	17.3205i	0.609711	+	1.05605i	−0.381577	+	0.924337i	$0.624619\pi$
$270$		0			0
$271$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$271$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$272$			8.00000i			0.485071i
$273$		0			0
$274$	−14.0000			−0.845771
$275$		0			0
$276$		0			0
$277$	−24.2487	+	14.0000i	−1.45696	+	0.841178i	−0.998861	−	0.0477206i	$-0.984804\pi$
$277$	−24.2487	+	14.0000i	−1.45696	+	0.841178i	−0.458103	+	0.888899i	$0.651471\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	11.0000	−	11.0000i	0.656205	−	0.656205i	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	11.0000	−	11.0000i	0.656205	−	0.656205i	0.954480	+	0.298275i	$0.0964112\pi$
$282$		0			0
$283$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$283$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−16.3923	+	4.39230i	−0.965926	+	0.258819i
$289$	−6.50000	+	11.2583i	−0.382353	+	0.662255i
$290$	−20.7846	+	12.0000i	−1.22051	+	0.704664i
$291$		0			0
$292$	8.05256	−	30.0526i	0.471240	−	1.75869i
$293$	25.9545	+	6.95448i	1.51628	+	0.406285i	0.918514	−	0.395388i	$-0.129390\pi$
$293$	25.9545	+	6.95448i	1.51628	+	0.406285i	0.597763	+	0.801673i	$0.296056\pi$
$294$		0			0
$295$		0			0
$296$	−17.3205	−	10.0000i	−1.00673	−	0.581238i
$297$		0			0
$298$			34.0000i			1.96957i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	10.9808	−	40.9808i	0.628757	−	2.34655i
$306$	8.19615	+	2.19615i	0.468543	+	0.125546i
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−24.0000			−1.35656			−0.678280	−	0.734803i	$-0.737274\pi$
$313$	−24.0000			−1.35656			−0.678280	+	0.734803i	$0.737274\pi$
$314$	16.3923	−	4.39230i	0.925071	−	0.247872i
$315$		0			0
$316$		0			0
$317$	3.00000	+	3.00000i	0.168497	+	0.168497i	0.786318	−	0.617822i	$-0.211985\pi$
$317$	3.00000	+	3.00000i	0.168497	+	0.168497i	−0.617822	+	0.786318i	$0.711985\pi$
$318$		0			0
$319$		0			0
$320$	24.0000	−	24.0000i	1.34164	−	1.34164i
$321$		0			0
$322$		0			0
$323$		0			0
$324$			18.0000i			1.00000i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−2.00000	+	3.46410i	−0.110432	+	0.191273i
$329$		0			0
$330$		0			0
$331$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$331$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$332$		0			0
$333$	−15.0000	+	15.0000i	−0.821995	+	0.821995i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	18.0000i		−	0.980522i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$		−	18.0000i		−	0.980522i	0.871576	−	0.490261i	$-0.163099\pi$
$338$		0			0
$339$		0			0
$340$	−16.3923	+	4.39230i	−0.888998	+	0.238206i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	4.00000	−	4.00000i	0.215041	−	0.215041i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	8.41858	+	31.4186i	0.450636	+	1.68180i	0.700609	+	0.713545i	$0.252912\pi$
$349$	8.41858	+	31.4186i	0.450636	+	1.68180i	−0.249973	+	0.968253i	$0.580422\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	34.1506	−	9.15064i	1.81765	−	0.487039i	0.821160	−	0.570697i	$-0.193327\pi$
$353$	34.1506	−	9.15064i	1.81765	−	0.487039i	0.996495	+	0.0836583i	$0.0266604\pi$
$354$		0			0
$355$		0			0
$356$	−6.00000	−	6.00000i	−0.317999	−	0.317999i
$357$		0			0
$358$		0			0
$359$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$359$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$360$	−18.0000	−	31.1769i	−0.948683	−	1.64317i
$361$	−16.4545	−	9.50000i	−0.866025	−	0.500000i
$362$	−6.58846	−	24.5885i	−0.346282	−	1.29234i
$363$		0			0
$364$		0			0
$365$	66.0000			3.45460
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$	3.00000	+	3.00000i	0.156174	+	0.156174i
$370$	10.9808	−	40.9808i	0.570863	−	2.13049i
$371$		0			0
$372$		0			0
$373$	−18.0000	−	31.1769i	−0.932005	−	1.61428i	−0.779890	−	0.625917i	$-0.784725\pi$
$373$	−18.0000	−	31.1769i	−0.932005	−	1.61428i	−0.152115	−	0.988363i	$-0.548608\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$379$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$383$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$384$		0			0
$385$		0			0
$386$	−5.00000	−	8.66025i	−0.254493	−	0.440795i
$387$		0			0
$388$	3.66025	+	13.6603i	0.185821	+	0.693494i
$389$			34.0000i			1.72387i	0.507020	+	0.861934i	$0.330747\pi$
$389$			34.0000i			1.72387i	−0.507020	+	0.861934i	$0.669253\pi$
$390$		0			0
$391$		0			0
$392$	19.1244	−	5.12436i	0.965926	−	0.258819i
$393$		0			0
$394$	25.9808	−	15.0000i	1.30889	−	0.755689i
$395$		0			0
$396$		0			0
$397$	−34.1506	−	9.15064i	−1.71397	−	0.459257i	−0.737579	−	0.675261i	$-0.764031\pi$
$397$	−34.1506	−	9.15064i	−1.71397	−	0.459257i	−0.976392	+	0.216004i	$0.930698\pi$
$398$		0			0
$399$		0			0
$400$	45.0333	+	26.0000i	2.25167	+	1.30000i
$401$	7.68653	+	28.6865i	0.383847	+	1.43254i	0.839976	+	0.542623i	$0.182569\pi$
$401$	7.68653	+	28.6865i	0.383847	+	1.43254i	−0.456129	+	0.889914i	$0.650764\pi$
$402$		0			0
$403$		0			0
$404$	−40.0000			−1.99007
$405$	−36.8827	+	9.88269i	−1.83272	+	0.491075i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−6.22243	+	23.2224i	−0.307679	+	1.14828i	0.622935	+	0.782274i	$0.285940\pi$
$409$	−6.22243	+	23.2224i	−0.307679	+	1.14828i	−0.930614	+	0.366002i	$0.880726\pi$
$410$	−8.19615	−	2.19615i	−0.404779	−	0.108460i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	29.0000	+	29.0000i	1.41337	+	1.41337i	0.731055	+	0.682318i	$0.239028\pi$
$421$	29.0000	+	29.0000i	1.41337	+	1.41337i	0.682318	+	0.731055i	$0.260972\pi$
$422$		0			0
$423$		0			0
$424$	−28.0000	+	28.0000i	−1.35980	+	1.35980i
$425$	−13.0000	−	22.5167i	−0.630593	−	1.09222i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$431$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$432$		0			0
$433$	20.7846	−	12.0000i	0.998845	−	0.576683i	0.0909384	−	0.995857i	$-0.471013\pi$
$433$	20.7846	−	12.0000i	0.998845	−	0.576683i	0.907906	+	0.419173i	$0.137680\pi$
$434$		0			0
$435$		0			0
$436$	19.1244	+	5.12436i	0.915891	+	0.245412i
$437$		0			0
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$		−	21.0000i		−	1.00000i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	9.00000	−	15.5885i	0.426641	−	0.738964i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	36.8827	+	9.88269i	1.74060	+	0.466393i	0.982581	−	0.185837i	$-0.0594997\pi$
$449$	36.8827	+	9.88269i	1.74060	+	0.466393i	0.758021	+	0.652230i	$0.226166\pi$
$450$	39.0000	−	39.0000i	1.83848	−	1.83848i
$451$		0			0
$452$	27.7128	+	16.0000i	1.30350	+	0.752577i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	34.1506	−	9.15064i	1.59750	−	0.428049i	0.653213	−	0.757174i	$-0.273421\pi$
$457$	34.1506	−	9.15064i	1.59750	−	0.428049i	0.944286	+	0.329125i	$0.106754\pi$
$458$	17.0000	−	29.4449i	0.794358	−	1.37587i
$459$		0			0
$460$		0			0
$461$	3.29423	−	12.2942i	0.153428	−	0.572599i	−0.845807	−	0.533488i	$-0.820881\pi$
$461$	3.29423	−	12.2942i	0.153428	−	0.572599i	0.999235	−	0.0391109i	$-0.0124526\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$	8.00000	+	13.8564i	0.371391	+	0.643268i
$465$		0			0
$466$	−5.85641	−	21.8564i	−0.271293	−	1.01248i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	21.0000	+	36.3731i	0.961524	+	1.66541i
$478$		0			0
$479$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$479$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$480$		0			0
$481$		0			0
$482$	38.0000			1.73085
$483$		0			0
$484$	−11.0000	+	19.0526i	−0.500000	+	0.866025i
$485$	−25.9808	+	15.0000i	−1.17973	+	0.681115i
$486$		0			0
$487$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$487$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$488$	−27.3205	−	7.32051i	−1.23674	−	0.331384i
$489$		0			0
$490$	21.0000	+	36.3731i	0.948683	+	1.64317i
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$		0			0
$493$		−	8.00000i		−	0.360302i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$499$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$500$	−17.5692	+	65.5692i	−0.785719	+	2.93234i
$501$		0			0
$502$		0			0
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$	−21.9615	−	81.9615i	−0.977275	−	3.64724i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−36.8827	+	9.88269i	−1.63480	+	0.438042i	−0.955300	−	0.295637i	$-0.904468\pi$
$509$	−36.8827	+	9.88269i	−1.63480	+	0.438042i	−0.679496	+	0.733679i	$0.737801\pi$
$510$		0			0
$511$		0			0
$512$	−16.0000	−	16.0000i	−0.707107	−	0.707107i
$513$		0			0
$514$	43.7128	+	11.7128i	1.92809	+	0.516630i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−40.0000			−1.75243			−0.876216	−	0.481919i	$-0.839940\pi$
$521$	−40.0000			−1.75243			−0.876216	+	0.481919i	$0.839940\pi$
$522$	16.3923	−	4.39230i	0.717472	−	0.192246i
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$	−72.7461	−	42.0000i	−3.15989	−	1.82436i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	20.0000	+	20.0000i	0.862261	+	0.862261i
$539$		0			0
$540$		0			0
$541$	11.0000	−	11.0000i	0.472927	−	0.472927i	−0.429934	−	0.902861i	$-0.641463\pi$
$541$	11.0000	−	11.0000i	0.472927	−	0.472927i	0.902861	+	0.429934i	$0.141463\pi$
$542$		0			0
$543$		0			0
$544$	2.92820	+	10.9282i	0.125546	+	0.468543i
$545$			42.0000i			1.79908i
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	−19.1244	+	5.12436i	−0.816952	+	0.218902i
$549$	−15.0000	+	25.9808i	−0.640184	+	1.10883i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−28.0000	+	28.0000i	−1.18961	+	1.18961i
$555$		0			0
$556$		0			0
$557$	−1.83013	−	6.83013i	−0.0775450	−	0.289402i	0.916253	−	0.400599i	$-0.131198\pi$
$557$	−1.83013	−	6.83013i	−0.0775450	−	0.289402i	−0.993798	+	0.111198i	$0.964531\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	11.0000	−	19.0526i	0.464007	−	0.803684i
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$	−17.5692	+	65.5692i	−0.739143	+	2.75852i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	34.6410	+	20.0000i	1.45223	+	0.838444i	0.998608	−	0.0527519i	$-0.0167993\pi$
$569$	34.6410	+	20.0000i	1.45223	+	0.838444i	0.453619	+	0.891196i	$0.350133\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−20.7846	+	12.0000i	−0.866025	+	0.500000i
$577$	−23.0000	−	23.0000i	−0.957503	−	0.957503i	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	−23.0000	−	23.0000i	−0.957503	−	0.957503i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	−4.75833	+	17.7583i	−0.197920	+	0.738649i
$579$		0			0
$580$	−24.0000	+	24.0000i	−0.996546	+	0.996546i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		−	44.0000i		−	1.82073i
$585$		0			0
$586$	38.0000			1.56977
$587$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$587$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−27.3205	−	7.32051i	−1.12287	−	0.300871i
$593$	−15.0000	+	15.0000i	−0.615976	+	0.615976i	−0.944497	−	0.328521i	$-0.893450\pi$
$593$	−15.0000	+	15.0000i	−0.615976	+	0.615976i	0.328521	+	0.944497i	$0.393450\pi$
$594$		0			0
$595$		0			0
$596$	12.4449	+	46.4449i	0.509761	+	1.90246i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	24.0000	−	41.5692i	0.978980	−	1.69564i	0.312861	−	0.949799i	$-0.398713\pi$
$601$	24.0000	−	41.5692i	0.978980	−	1.69564i	0.666120	−	0.745845i	$-0.267954\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−45.0788	−	12.0788i	−1.83272	−	0.491075i
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$		−	60.0000i		−	2.42933i
$611$		0			0
$612$	12.0000			0.485071
$613$	−1.36603	+	0.366025i	−0.0551732	+	0.0147836i	−0.286300	−	0.958140i	$-0.592425\pi$
$613$	−1.36603	+	0.366025i	−0.0551732	+	0.0147836i	0.231127	+	0.972924i	$0.425759\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	12.8109	−	47.8109i	0.515747	−	1.92479i	0.175382	−	0.984500i	$-0.443884\pi$
$617$	12.8109	−	47.8109i	0.515747	−	1.92479i	0.340365	−	0.940294i	$-0.389449\pi$
$618$		0			0
$619$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$619$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−79.0000			−3.16000
$626$	−32.7846	+	8.78461i	−1.31034	+	0.351104i
$627$		0			0
$628$	20.7846	−	12.0000i	0.829396	−	0.478852i
$629$	10.0000	+	10.0000i	0.398726	+	0.398726i
$630$		0			0
$631$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$631$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$632$		0			0
$633$		0			0
$634$	5.19615	+	3.00000i	0.206366	+	0.119145i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	24.0000	−	41.5692i	0.948683	−	1.64317i
$641$	43.3013	−	25.0000i	1.71030	−	0.987441i	0.776153	−	0.630544i	$-0.217168\pi$
$641$	43.3013	−	25.0000i	1.71030	−	0.987441i	0.934144	−	0.356897i	$-0.116165\pi$
$642$		0			0
$643$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$643$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$	6.58846	+	24.5885i	0.258819	+	0.965926i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−22.0000	+	38.1051i	−0.860927	+	1.49117i	0.0101092	+	0.999949i	$0.496782\pi$
$653$	−22.0000	+	38.1051i	−0.860927	+	1.49117i	−0.871036	+	0.491220i	$0.836551\pi$
$654$		0			0
$655$		0			0
$656$	−1.46410	+	5.46410i	−0.0571636	+	0.213337i
$657$	−45.0788	−	12.0788i	−1.75869	−	0.471240i
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$	−11.3468	−	42.3468i	−0.441339	−	1.64710i	−0.725426	−	0.688301i	$-0.758357\pi$
$661$	−11.3468	−	42.3468i	−0.441339	−	1.64710i	0.284087	−	0.958799i	$-0.408310\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−15.0000	+	25.9808i	−0.581238	+	1.00673i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−39.8372	−	23.0000i	−1.53561	−	0.886585i	−0.999088	−	0.0426985i	$-0.986405\pi$
$673$	−39.8372	−	23.0000i	−1.53561	−	0.886585i	−0.536522	−	0.843886i	$-0.680262\pi$
$674$	−6.58846	−	24.5885i	−0.253778	−	0.947112i
$675$		0			0
$676$		0			0
$677$	2.00000			0.0768662			0.0384331	−	0.999261i	$-0.487763\pi$
$677$	2.00000			0.0768662			0.0384331	+	0.999261i	$0.487763\pi$
$678$		0			0
$679$		0			0
$680$	−20.7846	+	12.0000i	−0.797053	+	0.460179i
$681$		0			0
$682$		0			0
$683$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$683$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$684$		0			0
$685$	−21.0000	−	36.3731i	−0.802369	−	1.38974i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$691$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$692$	4.00000	−	6.92820i	0.152057	−	0.263371i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	2.00000	−	2.00000i	0.0757554	−	0.0757554i
$698$	23.0000	+	39.8372i	0.870563	+	1.50786i
$699$		0			0
$700$		0			0
$701$			10.0000i			0.377695i	0.982006	+	0.188847i	$0.0604752\pi$
$701$			10.0000i			0.377695i	−0.982006	+	0.188847i	$0.939525\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	43.3013	−	25.0000i	1.62966	−	0.940887i
$707$		0			0
$708$		0			0
$709$	−9.56218	−	2.56218i	−0.359115	−	0.0962246i	0.0747503	−	0.997202i	$-0.476184\pi$
$709$	−9.56218	−	2.56218i	−0.359115	−	0.0962246i	−0.433865	+	0.900978i	$0.642851\pi$
$710$		0			0
$711$		0			0
$712$	−10.3923	−	6.00000i	−0.389468	−	0.224860i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$	−36.0000	−	36.0000i	−1.34164	−	1.34164i
$721$		0			0
$722$	−25.9545	−	6.95448i	−0.965926	−	0.258819i
$723$		0			0
$724$	−18.0000	−	31.1769i	−0.668965	−	1.15868i
$725$	−45.0333	−	26.0000i	−1.67250	−	0.965616i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$	90.1577	−	24.1577i	3.33689	−	0.894116i
$731$		0			0
$732$		0			0
$733$	29.0000	+	29.0000i	1.07114	+	1.07114i	0.997268	+	0.0738717i	$0.0235355\pi$
$733$	29.0000	+	29.0000i	1.07114	+	1.07114i	0.0738717	+	0.997268i	$0.476464\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	5.19615	+	3.00000i	0.191273	+	0.110432i
$739$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$739$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$740$		−	60.0000i		−	2.20564i
$741$		0			0
$742$		0			0
$743$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$743$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$744$		0			0
$745$	−88.3346	+	51.0000i	−3.23633	+	1.86850i
$746$	−36.0000	−	36.0000i	−1.31805	−	1.31805i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$751$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−9.00000	+	15.5885i	−0.327111	+	0.566572i	−0.981937	−	0.189207i	$-0.939408\pi$
$757$	−9.00000	+	15.5885i	−0.327111	+	0.566572i	0.654827	+	0.755779i	$0.272742\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.36603	+	0.366025i	0.0495184	+	0.0132684i	0.283493	−	0.958974i	$-0.408507\pi$
$761$	1.36603	+	0.366025i	0.0495184	+	0.0132684i	−0.233975	+	0.972243i	$0.575173\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	6.58846	+	24.5885i	0.238206	+	0.888998i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−50.5429	+	13.5429i	−1.82263	+	0.488371i	−0.997107	−	0.0760054i	$-0.975783\pi$
$769$	−50.5429	+	13.5429i	−1.82263	+	0.488371i	−0.825518	+	0.564376i	$0.809117\pi$
$770$		0			0
$771$		0			0
$772$	−10.0000	−	10.0000i	−0.359908	−	0.359908i
$773$	−1.83013	+	6.83013i	−0.0658251	+	0.245663i	−0.990997	−	0.133887i	$-0.957254\pi$
$773$	−1.83013	+	6.83013i	−0.0658251	+	0.245663i	0.925172	+	0.379549i	$0.123921\pi$
$774$		0			0
$775$		0			0
$776$	10.0000	+	17.3205i	0.358979	+	0.621770i
$777$		0			0
$778$	12.4449	+	46.4449i	0.446170	+	1.66513i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	24.2487	−	14.0000i	0.866025	−	0.500000i
$785$	36.0000	+	36.0000i	1.28490	+	1.28490i
$786$		0			0
$787$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$787$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$788$	30.0000	−	30.0000i	1.06871	−	1.06871i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−50.0000			−1.77443
$795$		0			0
$796$		0			0
$797$	−19.0526	+	11.0000i	−0.674876	+	0.389640i	−0.797922	−	0.602761i	$-0.794067\pi$
$797$	−19.0526	+	11.0000i	−0.674876	+	0.389640i	0.123045	+	0.992401i	$0.460734\pi$
$798$		0			0
$799$		0			0
$800$	71.0333	+	19.0333i	2.51141	+	0.672930i
$801$	−9.00000	+	9.00000i	−0.317999	+	0.317999i
$802$	21.0000	+	36.3731i	0.741536	+	1.28438i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−54.6410	+	14.6410i	−1.92226	+	0.515069i
$809$	−28.0000	+	48.4974i	−0.984428	+	1.70508i	−0.339975	+	0.940435i	$0.610418\pi$
$809$	−28.0000	+	48.4974i	−0.984428	+	1.70508i	−0.644453	+	0.764644i	$0.722915\pi$
$810$	−46.7654	+	27.0000i	−1.64317	+	0.948683i
$811$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$811$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$			34.0000i			1.18878i
$819$		0			0
$820$	−12.0000			−0.419058
$821$	−15.0263	+	4.02628i	−0.524421	+	0.140518i	−0.511311	−	0.859396i	$-0.670840\pi$
$821$	−15.0263	+	4.02628i	−0.524421	+	0.140518i	−0.0131101	+	0.999914i	$0.504173\pi$
$822$		0			0
$823$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$823$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$	−17.3205	−	10.0000i	−0.601566	−	0.347314i	0.168091	−	0.985771i	$-0.446240\pi$
$829$	−17.3205	−	10.0000i	−0.601566	−	0.347314i	−0.769657	+	0.638457i	$0.779573\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−14.0000			−0.485071
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$839$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$840$		0			0
$841$	6.50000	+	11.2583i	0.224138	+	0.388218i
$842$	50.2295	+	29.0000i	1.73102	+	0.999406i
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−28.0000	+	48.4974i	−0.961524	+	1.66541i
$849$		0			0
$850$	−26.0000	−	26.0000i	−0.891793	−	0.891793i
$851$		0			0
$852$		0			0
$853$	−41.0000	+	41.0000i	−1.40381	+	1.40381i	−0.616308	+	0.787505i	$0.711372\pi$
$853$	−41.0000	+	41.0000i	−1.40381	+	1.40381i	−0.787505	+	0.616308i	$0.788628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			8.00000i			0.273275i	0.990621	+	0.136637i	$0.0436295\pi$
$857$			8.00000i			0.273275i	−0.990621	+	0.136637i	$0.956370\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$	16.3923	+	4.39230i	0.557355	+	0.149343i
$866$	24.0000	−	24.0000i	0.815553	−	0.815553i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	28.0000			0.948200
$873$	20.4904	−	5.49038i	0.693494	−	0.185821i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	12.8109	−	47.8109i	0.432593	−	1.61446i	−0.314169	−	0.949367i	$-0.601726\pi$
$877$	12.8109	−	47.8109i	0.432593	−	1.61446i	0.746762	−	0.665092i	$-0.231608\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	27.7128	+	16.0000i	0.933668	+	0.539054i	0.887970	−	0.459902i	$-0.152115\pi$
$881$	27.7128	+	16.0000i	0.933668	+	0.539054i	0.0456985	+	0.998955i	$0.485449\pi$
$882$	−7.68653	−	28.6865i	−0.258819	−	0.965926i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$		0			0
$890$	6.58846	−	24.5885i	0.220846	−	0.824207i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	54.0000			1.80200
$899$		0			0
$900$	39.0000	−	67.5500i	1.30000	−	2.25167i
$901$	24.2487	−	14.0000i	0.807842	−	0.466408i
$902$		0			0
$903$		0			0
$904$	43.7128	+	11.7128i	1.45387	+	0.389562i
$905$	54.0000	−	54.0000i	1.79502	−	1.79502i
$906$		0			0
$907$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$907$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$908$		0			0
$909$			60.0000i			1.99007i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$	43.3013	−	25.0000i	1.43228	−	0.826927i
$915$		0			0
$916$	12.4449	−	46.4449i	0.411190	−	1.53458i
$917$		0			0
$918$		0			0
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$		0			0
$921$		0			0
$922$		−	18.0000i		−	0.592798i
$923$		0			0
$924$		0			0
$925$	88.7917	−	23.7917i	2.91945	−	0.782265i
$926$		0			0
$927$		0			0
$928$	16.0000	+	16.0000i	0.525226	+	0.525226i
$929$	−15.7391	+	58.7391i	−0.516383	+	1.92717i	−0.190965	+	0.981597i	$0.561162\pi$
$929$	−15.7391	+	58.7391i	−0.516383	+	1.92717i	−0.325418	+	0.945570i	$0.605505\pi$
$930$		0			0
$931$		0			0
$932$	−16.0000	−	27.7128i	−0.524097	−	0.907763i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	38.0000			1.24141			0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000			1.24141			0.620703	+	0.784046i	$0.286847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−19.0000	−	19.0000i	−0.619382	−	0.619382i	0.325991	−	0.945373i	$-0.394302\pi$
$941$	−19.0000	−	19.0000i	−0.619382	−	0.619382i	−0.945373	+	0.325991i	$0.894302\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$947$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	48.4974	−	28.0000i	1.57099	−	0.907009i	0.574937	−	0.818198i	$-0.305026\pi$
$953$	48.4974	−	28.0000i	1.57099	−	0.907009i	0.996048	−	0.0888114i	$-0.0283068\pi$
$954$	42.0000	+	42.0000i	1.35980	+	1.35980i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$			31.0000i			1.00000i
$962$		0			0
$963$		0			0
$964$	51.9090	−	13.9090i	1.67188	−	0.447978i
$965$	15.0000	−	25.9808i	0.482867	−	0.836350i
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	−8.05256	+	30.0526i	−0.258819	+	0.965926i
$969$		0			0
$970$	−30.0000	+	30.0000i	−0.963242	+	0.963242i
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−40.0000			−1.28037
$977$	−36.8827	+	9.88269i	−1.17998	+	0.316175i	−0.794919	−	0.606715i	$-0.792487\pi$
$977$	−36.8827	+	9.88269i	−1.17998	+	0.316175i	−0.385063	+	0.922890i	$0.625820\pi$
$978$		0			0
$979$		0			0
$980$	42.0000	+	42.0000i	1.34164	+	1.34164i
$981$	7.68653	−	28.6865i	0.245412	−	0.915891i
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$	77.9423	+	45.0000i	2.48345	+	1.43382i
$986$	−2.92820	−	10.9282i	−0.0932530	−	0.348025i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−31.0000	−	53.6936i	−0.981780	−	1.70049i	−0.655454	−	0.755235i	$-0.727523\pi$
$997$	−31.0000	−	53.6936i	−0.981780	−	1.70049i	−0.326326	−	0.945257i	$-0.605811\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	676.2.l.g.319.1		4
4.3	odd	2	CM	676.2.l.g.319.1		4
13.2	odd	12	inner	676.2.l.g.587.1		4
13.3	even	3	inner	676.2.l.g.427.1		4
13.4	even	6		52.2.f.a.47.1	yes	2
13.5	odd	4	inner	676.2.l.g.19.1		4
13.6	odd	12		676.2.f.b.239.1		2
13.7	odd	12		52.2.f.a.31.1	✓	2
13.8	odd	4		676.2.l.a.19.1		4
13.9	even	3		676.2.f.b.99.1		2
13.10	even	6		676.2.l.a.427.1		4
13.11	odd	12		676.2.l.a.587.1		4
13.12	even	2		676.2.l.a.319.1		4
39.17	odd	6		468.2.n.c.307.1		2
39.20	even	12		468.2.n.c.343.1		2
52.3	odd	6	inner	676.2.l.g.427.1		4
52.7	even	12		52.2.f.a.31.1	✓	2
52.11	even	12		676.2.l.a.587.1		4
52.15	even	12	inner	676.2.l.g.587.1		4
52.19	even	12		676.2.f.b.239.1		2
52.23	odd	6		676.2.l.a.427.1		4
52.31	even	4	inner	676.2.l.g.19.1		4
52.35	odd	6		676.2.f.b.99.1		2
52.43	odd	6		52.2.f.a.47.1	yes	2
52.47	even	4		676.2.l.a.19.1		4
52.51	odd	2		676.2.l.a.319.1		4
104.43	odd	6		832.2.k.d.255.1		2
104.59	even	12		832.2.k.d.447.1		2
104.69	even	6		832.2.k.d.255.1		2
104.85	odd	12		832.2.k.d.447.1		2
156.59	odd	12		468.2.n.c.343.1		2
156.95	even	6		468.2.n.c.307.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.2.f.a.31.1	✓	2	13.7	odd	12
52.2.f.a.31.1	✓	2	52.7	even	12
52.2.f.a.47.1	yes	2	13.4	even	6
52.2.f.a.47.1	yes	2	52.43	odd	6
468.2.n.c.307.1		2	39.17	odd	6
468.2.n.c.307.1		2	156.95	even	6
468.2.n.c.343.1		2	39.20	even	12
468.2.n.c.343.1		2	156.59	odd	12
676.2.f.b.99.1		2	13.9	even	3
676.2.f.b.99.1		2	52.35	odd	6
676.2.f.b.239.1		2	13.6	odd	12
676.2.f.b.239.1		2	52.19	even	12
676.2.l.a.19.1		4	13.8	odd	4
676.2.l.a.19.1		4	52.47	even	4
676.2.l.a.319.1		4	13.12	even	2
676.2.l.a.319.1		4	52.51	odd	2
676.2.l.a.427.1		4	13.10	even	6
676.2.l.a.427.1		4	52.23	odd	6
676.2.l.a.587.1		4	13.11	odd	12
676.2.l.a.587.1		4	52.11	even	12
676.2.l.g.19.1		4	13.5	odd	4	inner
676.2.l.g.19.1		4	52.31	even	4	inner
676.2.l.g.319.1		4	1.1	even	1	trivial
676.2.l.g.319.1		4	4.3	odd	2	CM
676.2.l.g.427.1		4	13.3	even	3	inner
676.2.l.g.427.1		4	52.3	odd	6	inner
676.2.l.g.587.1		4	13.2	odd	12	inner
676.2.l.g.587.1		4	52.15	even	12	inner
832.2.k.d.255.1		2	104.43	odd	6
832.2.k.d.255.1		2	104.69	even	6
832.2.k.d.447.1		2	104.59	even	12
832.2.k.d.447.1		2	104.85	odd	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 676 = 2^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	676.l (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(1.73205 - 1.00000i) q^{4} +(3.00000 + 3.00000i) q^{5} +(2.00000 - 2.00000i) q^{8} +(-1.50000 - 2.59808i) q^{9} +(5.19615 + 3.00000i) q^{10} +(2.00000 - 3.46410i) q^{16} +(-1.73205 + 1.00000i) q^{17} +(-3.00000 - 3.00000i) q^{18} +(8.19615 + 2.19615i) q^{20} +13.0000i q^{25} +(-2.00000 + 3.46410i) q^{29} +(1.46410 - 5.46410i) q^{32} +(-2.00000 + 2.00000i) q^{34} +(-5.19615 - 3.00000i) q^{36} +(-1.83013 - 6.83013i) q^{37} +12.0000 q^{40} +(-1.36603 + 0.366025i) q^{41} +(3.29423 - 12.2942i) q^{45} +(6.06218 + 3.50000i) q^{49} +(4.75833 + 17.7583i) q^{50} -14.0000 q^{53} +(-1.46410 + 5.46410i) q^{58} +(-5.00000 - 8.66025i) q^{61} -8.00000i q^{64} +(-2.00000 + 3.46410i) q^{68} +(-8.19615 - 2.19615i) q^{72} +(11.0000 - 11.0000i) q^{73} +(-5.00000 - 8.66025i) q^{74} +(16.3923 - 4.39230i) q^{80} +(-4.50000 + 7.79423i) q^{81} +(-1.73205 + 1.00000i) q^{82} +(-8.19615 - 2.19615i) q^{85} +(-1.09808 - 4.09808i) q^{89} -18.0000i q^{90} +(-1.83013 + 6.83013i) q^{97} +(9.56218 + 2.56218i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 12 q^{5} + 8 q^{8} - 6 q^{9} + 8 q^{16} - 12 q^{18} + 12 q^{20} - 8 q^{29} - 8 q^{32} - 8 q^{34} + 10 q^{37} + 48 q^{40} - 2 q^{41} - 18 q^{45} - 26 q^{50} - 56 q^{53} + 8 q^{58} - 20 q^{61}+ \cdots + 14 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 12 * q^5 + 8 * q^8 - 6 * q^9 + 8 * q^16 - 12 * q^18 + 12 * q^20 - 8 * q^29 - 8 * q^32 - 8 * q^34 + 10 * q^37 + 48 * q^40 - 2 * q^41 - 18 * q^45 - 26 * q^50 - 56 * q^53 + 8 * q^58 - 20 * q^61 - 8 * q^68 - 12 * q^72 + 44 * q^73 - 20 * q^74 + 24 * q^80 - 18 * q^81 - 12 * q^85 + 6 * q^89 + 10 * q^97 + 14 * q^98

Introduction

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